Signals and Systems: end-semester study guide
Course: EC 203 Signals and Systems · Exam: End-semester, 2 Dec 2026 · Author: Prof. Lakshmi Varadarajan · Date: 18 Nov 2026
Tip
Make your own copy with File → Make a copy, then tick the self-check list as you revise. Whatever is still unticked three days before the exam is your plan for those days.
The course on one page
| Unit | What it covers | Remember |
|---|---|---|
| 1. Signals | Continuous and discrete signals, even and odd parts, energy and power | A signal has finite energy or finite average power, never both nonzero |
| 2. LTI systems | Linearity, time invariance, causality, stability, impulse response | An LTI system is completely described by its impulse response |
| 3. Convolution | Continuous and discrete convolution, graphical method | Flip one signal, slide it, multiply and add |
| 4. Fourier | Fourier series, Fourier transform, frequency response | Convolution in time is multiplication in frequency |
| 5. Laplace | Transform, region of convergence, poles and zeros, solving ODEs | A causal system is stable when every pole is in the left half plane |
| 6. Sampling and Z | Sampling theorem, aliasing, Z-transform | Sample faster than twice the highest frequency |
Formula sheet
Convolution
y(t) = \int_{-\infty}^{\infty} x(\tau)\,h(t-\tau)\,d\tau, \qquad y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k]
Fourier transform
X(j\omega) = \int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt, \qquad x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)\,e^{j\omega t}\,d\omega
Parseval's relation:
\int_{-\infty}^{\infty} |x(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(j\omega)|^2\,d\omega
Laplace transform
X(s) = \int_{-\infty}^{\infty} x(t)\,e^{-st}\,dt, \qquad e^{-at}u(t) \;\longleftrightarrow\; \frac{1}{s+a}, \quad \text{Re}(s) > -a
Sampling and the Z-transform
f_s > 2 f_{\max}, \qquad X(z) = \sum_{n=-\infty}^{\infty} x[n]\,z^{-n}, \qquad a^n u[n] \;\longleftrightarrow\; \frac{1}{1 - a z^{-1}}, \quad |z| > |a|
Stability
\text{BIBO stable} \iff \int_{-\infty}^{\infty} |h(t)|\,dt < \infty
For a causal system with a rational transfer function: every pole of H(s) in the left half plane, or every pole of H(z) inside the unit circle.
Key diagrams
How the transforms relate:
flowchart LR
T["x(t) in time"] -->|"Laplace"| S["X(s)"]
S -->|"s = jω, if the ROC includes the jω axis"| F["X(jω)"]
T -->|"sample every T seconds"| N["x[n]"]
N -->|"Z-transform"| Z["X(z)"]
Z -->|"z = e^jω on the unit circle"| D["DTFT"]
Also be ready to sketch: the graphical convolution of two rectangular pulses (the result is a trapezoid, or a triangle when the pulses are equal), and the spectrum of a sampled signal showing copies at multiples of f_s overlapping when f_s is too low.
Common mistakes
| Mistake | Fix |
|---|---|
| Giving a Laplace or Z-transform without its region of convergence | Always state the ROC. Two different signals can share the same expression |
Mixing \omega in rad/s with f in Hz |
Remember \omega = 2\pi f, and check which one the question uses |
| Flipping the wrong signal in graphical convolution | Flip h, slide it across x, and label the overlap limits at each stage |
| Confusing the Nyquist rate with the Nyquist frequency | The Nyquist rate is 2 f_{\max}. The Nyquist frequency is f_s / 2 |
| Calling a system stable because its output looks bounded for one input | BIBO stability must hold for every bounded input. Check the impulse response |
Exam strategy
- Part A has ten short questions for 2 marks each. Spend no more than 25 minutes on it.
- Part B has five long questions for 16 marks each, with a choice in each. Start with convolution or Laplace, where the method earns most of the marks even if the arithmetic slips.
- Leave ten minutes to check every transform has its ROC.
Self-check
Where to look
- Units 1 and 2: Lectures 1 to 6 and Tutorial 1.
- Unit 3: Lectures 7 to 9 and Tutorial 2.
- Units 4 and 5: Lectures 10 to 16, Tutorials 3 to 5 and the 2025 end-semester paper.
- Unit 6: Lectures 17 to 22 and Tutorial 6.